Shape invariance and the exactness of the quantum Hamilton–Jacobi formalism (original) (raw)
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Shape invariance and the exactness of quantum Hamilton-Jacobi formalism
2007
Quantum Hamilton-Jacobi Theory and supersymmetric quantum mechanics (SUSYQM) are two parallel methods to determine the spectra of a quantum mechanical systems without solving the Schrödinger equation. It was recently shown that the shape invariance, which is an integrability condition in SUSYQM formalism, can be utilized to develop an iterative algorithm to determine the quantum momentum functions. In this paper, we show that shape invariance also suffices to determine the eigenvalues in Quantum Hamilton-Jacobi Theory. Quantum Hamilton-Jacobi Theory and supersymmetric quantum mechanics (SUSYQM) are two very different methods that give eigenvalues for quantum mechanical systems without solving the Schrödinger differential eigenvalue equation. Supersymmetric quantum mechanics is a generalization of Dirac's ladder operator method 4 for the harmonic oscillator. This method consists of factorizing Schrödinger's second order differential operator into two first order differential operators that play roles analogous to ladder operators. If the interaction of a quantum mechanical system is described by shape invariant potentials [1, 2], SUSYQM allows one to generate all eigenvalues and eigenfunctions through algebraic methods. Another formulation of quantum mechanics, the Quantum Hamilton-Jacobi (QHJ) formalism, was developed by Leacock and Padgett [3] and independently by Gozzi [4]. It was made popular by a series of papers by Kapoor et. al. [5]. In this formalism one works with the quantum momentum function (QMF) p(x), which is related to the wave function ψ through the relationship p(x) = −ψ′(x)/ψ(x), where prime denotes differentiation with respect to x. Our definition of QMF's differs by a factor of i ≡ √ −1 from that of ref. [3, 5, 6], where they define p(x) = −i ψ ′ (x) ψ(x) ; we use p(x) = − ψ ′ (x) ψ(x). It was shown, on a case by case basis, that the singularity structure of the function p(x) determines the eigenvalues of the Hamiltonian [3, 4, 5] for all known solvable potentials. Kapoor and his collaborators have shown that the QHJ formalism can be used not only to determine the eigenvalues of the Hamiltonian of the system, but also its eigenfunctions [7]. They have also used QHJ to analyze Quasi-exactly solvable systems where only an incomplete set of the eigenspectra can be derived analytically and also to study periodic potentials [8]. It is important to note that all cases worked out in Refs. [3, 5] satisfied the integrability condition known as the translational shape invariance for which the SUSYQM method always gave the exact result.
Exactly solvable systems and the quantum Hamilton–Jacobi formalism
Physics Letters A, 2005
We connect Quantum Hamilton-Jacobi Theory with supersymmetric quantum mechanics (SUSYQM). We show that the shape invariance, which is an integrability condition of SUSYQM, translates into fractional linear relations among the quantum momentum functions. Supersymmetric quantum mechanics (SUSYQM) has provided a powerful tool in analyzing the underlying structure of the Schrödinger equation [1]. In addition to connecting apparently distinct potentials, SUSYQM allows for algebraic solutions to a large class of such potentials: the known shape invariant potentials [2]. Another formulation of quantum mechanics, the Quantum Hamilton-Jacobi (QHJ) formalism, was developed by Leacock and Padgett [3] and independently by Gozzi [4]. In this formalism, which follows classical mechanics closely, one works with the quantum momentum function p(x), which is the quantum analog of the classical momentum function p c (x) = √ E − V . It was shown [3, 4] that the singularity structure of the function p(x) determines the eigenvalues of the Hamiltonian. Kapoor and his collaborators [5] have shown that the QHJ formalism can be used not only to determine the eigenvalues of the Hamiltonian of the system, but also its eigenfunctions.
Quantum Hamilton-Jacobi Quantization and Shape Invariance
Cornell University - arXiv, 2022
Quantum Hamilton-Jacobi quantization scheme uses the singularity structure of the potential of a quantum mechanical system to generate its eigenspectrum [1, 2] and eigenfunctions, and its efficacy has been demonstrated for many well known conventional potentials [3]. Using a recent work in supersymmetric quantum mechanics we prove that the solvability of all conventional potentials with the quantum Hamilton-Jacobi formalism follows from their shape invariance.
Quantum Hamilton-Jacobi Theory
Physical Review Letters, 2007
Quantum canonical transformations have attracted interest since the beginning of quantum theory. Based on their classical analogues, one would expect them to provide a powerful quantum tool. However, the difficulty of solving a nonlinear operator partial differential equation such as the quantum Hamilton-Jacobi equation (QHJE) has hindered progress along this otherwise promising avenue. We overcome this difficulty. We show that solutions to the QHJE can be constructed by a simple prescription starting from the propagator of the associated Schrödinger equation. Our result opens the possibility of practical use of quantum Hamilton-Jacobi theory. As an application we develop a surprising relation between operator ordering and the density of paths around a semiclassical trajectory.
The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics
The Journal of Geometric Mechanics, 2009
We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi (HJ) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the HJ theory on the tangent bundle of a configuration manifold, the quantum HJ theory, HJ problems for general differential operators and the HJ problem for Lie groups.
Canonical transformations and the Hamilton-Jacobi theory in quantum mechanics
Canadian Journal of Physics, 1999
Canonical transformations using the idea of quantum generating functions are applied to construct a quantum Hamilton-Jacobi theory, based on the analogy with the classical case. An operator and a c-number form of the time-dependent quantum Hamilton-Jacobi equation are derived and used to find dynamical solutions of quantum problems. The phase-space picture of quantum mechanics is discussed in connection with the present theory.PACS Nos.: 03.65-w, 03.65Ca, 03.65Ge
Journal of Mathematical Physics, 2013
Within the context of Supersymmetric Quantum Mechanics and its related hierarchies of integrable quantum Hamiltonians and potentials, a general programme is outlined and applied to its first two simplest illustrations. Going beyond the usual restriction of shape invariance for intertwined potentials, it is suggested to require a similar relation for Hamiltonians in the hierarchy separated by an arbitrary number of levels, N . By requiring further that these two Hamiltonians be in fact identical up to an overall shift in energy, a periodic structure is installed in the hierarchy of quantum systems which should allow for its solution. Specific classes of orthogonal polynomials characteristic of such periodic hierarchies are thereby generated, while the methods of Supersymmetric Quantum Mechanics then lead to generalised Rodrigues formulae and recursion relations for such polynomials. The approach also offers the practical prospect of quantum modelling through the engineering of quantum potentials from experimental energy spectra. In this paper these ideas are presented and solved explicitly for the cases N = 1 and N = 2. The latter case is related to the generalised Laguerre polynomials, for which indeed new results are thereby obtained. At the same time new classes of integrable quantum potentials which generalise that of the harmonic oscillator and which are characterised by two arbitrary energy gaps are identified, for which a complete solution is achieved algebraically.
Equilibrium positions and eigenfunctions of shape invariant ('discrete') quantum mechanics
2005
Certain aspects of the integrability/solvability of the Calogero-Sutherland-Moser systems and the Ruijsenaars-Schneider-van Diejen systems with rational and trigonometric potentials are reviewed. The equilibrium positions of classical multi-particle systems and the eigenfunctions of single-particle quantum mechanics are described by the same orthogonal polynomials: the Hermite, Laguerre, Jacobi, continuous Hahn, Wilson and Askey-Wilson polynomials. The Hamiltonians of these single-particle quantum mechanical systems have two remarkable properties, factorization and shape invariance.
Hamilton-Jacobi/action-angle quantum mechanics
Physical Review D, 1983
A quantum mechanics is constructed which is patterned on classical Hamilton-Jacobi theory. The dynamical basis of the theory is a quantum Hamilton-Jacobi equation with accompanying physical boundary conditions. Basic objects of the formulation are quantum Hamilton's principal and characteristic functions. As an application of the formalisxn a theory of quantum action-angle variables is set up. The central feature of this theory is the definition of the quantum action variable which permits the determination of the bound-state energy levels without solving the dynamical equation.
New exactly solvable Hamiltonians: Shape invariance and self-similarity
Physical Review A, 1993
We discuss in some detail the self-similar potentials of Shabat and Spiridonov which are reflectionless and have an infinite number of bound states. We demonstrate that these self-similar potentials are in fact shape invariant potentials within the formalism of supersymmetric quantum mechanics. In particular, using a scaling ansatz for the change of parameters, we obtain a large class of new, reflectionless, shape invariant potentials of which the Shabat-Spiridonov ones are a special case.